Complexity Growth in Integrable and Chaotic Models

Defining quantum complexity has been a long standing problem in condensed matter physics, quantum information and more recently quantum gravity. Using gate complexity from the theory of computation, one can upper-bound the unitary evolution by computing the length of the geodesic generated by the Hamiltonian on the U(N) manifold. In this formalism, one can study a type of local obstruction to complexity growth that is the appearance of conjugate points along the geodesic. We show that perturbatively, in the free SYK_q=2 model, the conjugate point appears at the length of order of O(N^1/2), and that for an integrable interacting theory - a deformed free SYK_q=2, the complexity is upper bounded by O(poly(N)), and that in the chaotic SYK_q>2, the conjugate points move away to O(exp(N)). This matches up with the intuition that integrable theory should generate simple unitaries and chaotic Hamiltonians should generate complex unitary dynamics. To combine the complexity theory and the quantum thermalization, we explore the complexity of the eigenstate of free, integrable interacting and chaotic Hamiltonian and use the eigenstate thermalization hypothesis to predict the appearance of conjugate points in chaotic systems.